Subgroups — #17192019

Order 4352.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541773	C₂	1	1	yes	maximal, characteristic
2	#14531018	C₂	1	2	yes	maximal
3	#14531013	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530945	C₂	1	2	yes	maximal, complemented
5	#14530918	C₂	1	1	yes	maximal, characteristic
6	#14530846	C₂	1	2	yes	maximal
7	#14530834	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×#63214	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#10587500	C₂	1	2	yes	maximal, complemented
10	C₄×#10587499	C₂	1	1	yes	maximal, characteristic
11	C₁₇×#3383	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763532	—	2	1	no	—
13	C₂×#10587500	C₂²	1	2	yes	—
14	C₂×#10587499	C₂²	1	1	yes	characteristic
15	#11763193	C₂²	1	2	yes	—
16	#11763189	C₂²	1	1	yes	characteristic
17	#11763032	—	2	2	no	—
18	#11763015	—	2	1	no	—
19	C₄×Dic₆₈	C₂²	1	2	yes	—
20	C₄×D₁₃₆	C₂²	1	1	yes	characteristic
21	#11762636	C₂²	1	1	yes	characteristic
22	#11762634	C₂²	1	2	yes	—
23	C₄⋊#63214	C₂²	1	1	yes	characteristic
24	C₄⋊#63214	C₂²	1	2	yes	—
25	#11762613	C₂²	1	1	yes	characteristic
26	#11762612	C₂²	1	2	yes	—
27	#11762612	C₂²	1	2	yes	—
28	#11762612	C₂²	1	2	yes	—
29	#11762609	C₂²	1	2	yes	—
30	#11762609	C₂²	1	1	yes	characteristic
31	C₄×#63214	C₂²	1	1	yes	characteristic
32	C₄×#63214	C₂²	1	2	yes	—
33	C₁₇×#557	—	2	1	no	—
34	C₁₇×#500	C₂²	1	1	yes	characteristic
35	C₁₇×#499	C₂²	1	2	yes	—
36	C₁₇×#479	—	2	2	no	—
37	C₁₇×#475	—	2	1	no	—
38	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
41	C₂×Q₁₆×C₁₇	C₂³	1	1	yes	characteristic
42	C₁₇×C₄⋊Q₈	—	2	1	no	—
43	C₁₇×C₄⋊Q₈	—	4	1	no	—
44	C₁₇×#125	—	2	2	no	—
45	C₁₇×#125	—	4	2	no	—
46	C₄×Q₈×C₁₇	—	2	1	no	—
47	C₄×Q₈×C₁₇	—	2	2	no	—
48	C₄×Q₈×C₁₇	—	4	1	no	—
49	C₄×Q₈×C₁₇	D₄	1	1	yes	characteristic
50	#10587632	—	4	1	no	—
51	C₂×Dic₆₈	—	2	1	no	—
52	C₂×Dic₆₈	C₂³	1	2	yes	—
53	C₂×D₁₃₆	C₂³	1	1	yes	characteristic
54	C₂×#63219	—	2	2	no	—
55	#10587584	—	4	2	no	—
56	#10587578	—	4	1	no	—
57	#10587561	—	2	2	no	—
58	C₄×D₆₈	—	2	1	no	—
59	#10587555	—	2	1	no	—
60	C₄×Dic₃₄	—	2	2	no	—
61	C₁₇×#107	—	2	1	no	—
62	C₁₇×#107	C₂³	1	1	yes	characteristic
63	C₁₇×#107	C₂³	1	2	yes	—
64	C₁₇×#106	—	2	2	no	—
65	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
66	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
67	C₁₇×#103	—	2	2	no	—
68	C₁₇×#103	—	2	1	no	—
69	C₄×C₈×C₁₇	C₂³	1	2	yes	—
70	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	#10587500	—	2	4	no	—
72	#10587499	—	2	2	no	—
73	#10587492	—	2	1	no	—
74	#10587489	C₂³	1	1	yes	characteristic
75	#10587487	—	2	2	no	—
76	C₂×#63214	C₂³	1	1	yes	characteristic
77	C₂×#63214	C₂³	1	2	yes	—
78	C₂×#63214	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₇	—	2	1	no	—
80	C₂×Q₈×C₁₇	—	4	1	no	—
81	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
82	C₂×D₆₈	—	2	1	no	—
83	C₂×C₄×D₁₇	—	4	1	no	—
84	C₂×Dic₃₄	—	2	2	no	—
85	Q₁₆×C₁₇	—	2	1	no	—
86	Q₁₆×C₁₇	#41	1	2	yes	—
87	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₁₇	—	2	1	no	—
91	C₄⋊C₄×C₁₇	—	2	2	no	—
92	C₄⋊C₄×C₁₇	—	4	2	no	—
93	C₄⋊C₄×C₁₇	—	4	2	no	—
94	C₄⋊C₄×C₁₇	—	4	1	no	—
95	C₄⋊C₄×C₁₇	—	4	1	no	—
96	C₄⋊C₄×C₁₇	—	4	2	no	—
97	C₄⋊C₄×C₁₇	D₈	1	2	yes	—
98	C₄⋊C₄×C₁₇	D₈	1	1	yes	characteristic
99	C₄²×C₁₇	—	4	2	no	—
100	C₄²×C₁₇	—	4	1	no	—
101	C₄²×C₁₇	C₂×D₄	1	2	yes	—
102	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
103	#63227	—	4	1	no	—
104	#63227	—	4	2	no	—
105	#63227	—	4	2	no	—
106	C₄⋊Dic₁₇	—	2	1	no	—
107	C₄⋊Dic₁₇	—	4	1	no	—
108	C₄⋊Dic₁₇	—	4	1	no	—
109	#63225	—	4	2	no	—
110	C₄×Dic₁₇	—	4	2	no	—
111	Dic₆₈	—	2	2	no	—
112	Dic₆₈	—	2	2	no	—
113	Dic₆₈	—	8	1	no	—
114	D₁₃₆	—	2	1	no	—
115	D₁₃₆	—	2	1	no	—
116	#63219	—	8	2	no	—
117	#63214	—	2	1	no	—
118	#63214	—	2	2	no	—
119	#63214	—	2	1	no	—
120	#33931	—	17	1	no	maximal, Sylow
121	C₂²×D₁₇	—	4	1	no	—
122	Q₈×C₁₇	—	2	2	no	—
123	Q₈×C₁₇	—	2	2	no	—
124	Q₈×C₁₇	—	4	1	no	—
125	Q₈×C₁₇	—	8	2	no	—
126	C₂×C₄×C₁₇	—	4	1	no	—
127	C₂×C₄×C₁₇	—	4	1	no	—
128	C₂×C₄×C₁₇	—	4	2	no	—
129	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₁₇	—	4	2	no	—
135	C₂×Dic₁₇	—	4	1	no	—
136	D₆₈	—	4	1	no	—
137	D₆₈	—	4	1	no	—
138	C₄×D₁₇	—	8	2	no	—
139	Dic₃₄	—	4	2	no	—
140	Dic₃₄	—	4	2	no	—
141	C₈×C₁₇	—	2	2	no	—
142	C₈×C₁₇	—	2	1	no	—
143	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
144	C₈×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
145	#3393	—	17	1	no	—
146	#3383	D₁₇	1	1	yes	characteristic, complemented
147	#2242	—	17	2	no	—
148	#2241	—	17	1	no	—
149	#2229	—	17	2	no	—
150	#2224	—	17	1	no	—
151	#2211	—	17	2	no	—
152	#2209	—	17	1	no	—
153	Q₈×C₁₆	—	17	1	no	—
154	C₄×Q₃₂	—	17	2	no	—
155	C₄×SD₃₂	—	17	1	no	—
156	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₁₇	—	8	1	no	—
158	C₂×D₁₇	—	8	1	no	—
159	C₂×D₁₇	—	8	1	no	—
160	C₄×C₁₇	—	2	2	no	—
161	C₄×C₁₇	—	2	2	no	—
162	C₄×C₁₇	—	2	1	no	—
163	C₄×C₁₇	—	4	2	no	—
164	C₄×C₁₇	—	8	2	no	—
165	C₄×C₁₇	—	8	1	no	—
166	C₄×C₁₇	#508	1	2	yes	—
167	C₄×C₁₇	#548	1	1	yes	characteristic
168	C₄×C₁₇	#577	1	1	yes	characteristic
169	Dic₁₇	—	8	2	no	—
170	Dic₁₇	—	8	1	no	—
171	#557	—	2	1	no	—
172	#556	—	34	1	no	—
173	C₂×Q₃₂	—	17	2	no	—
174	C₂×SD₃₂	—	17	1	no	—
175	#500	C₂×D₁₇	1	1	yes	characteristic
176	#499	C₂×D₁₇	1	2	yes	—
177	#495	—	17	2	no	—
178	#494	—	17	1	no	—
179	#479	—	2	2	no	—
180	#475	—	2	1	no	—
181	#464	—	34	2	no	—
182	#461	—	34	1	no	—
183	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
184	C₄×Q₁₆	—	17	2	no	—
185	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
186	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
187	C₄×D₈	—	17	1	no	—
188	#367	—	17	1	no	—
189	#366	—	17	2	no	—
190	C₄⋊C₁₆	—	17	1	no	—
191	C₄⋊C₁₆	—	17	2	no	—
192	#358	—	17	1	no	—
193	#358	—	17	2	no	—
194	#358	—	17	2	no	—
195	#358	—	17	2	no	—
196	#357	—	17	1	no	—
197	#357	—	17	2	no	—
198	C₄×C₁₆	—	17	1	no	—
199	C₄×C₁₆	—	17	2	no	—
200	C₂×C₁₇	#3358	1	1	yes	characteristic
201	C₂×C₁₇	C₂×#508	1	1	yes	characteristic
202	C₂×C₁₇	#3417	1	1	yes	characteristic
203	D₁₇	—	16	1	no	—
204	C₂×Q₁₆	—	17	2	no	—
205	C₂×Q₁₆	—	34	1	no	—
206	C₂×Q₁₆	C₂²×D₁₇	1	1	yes	characteristic
207	C₂×SD₁₆	—	34	2	no	—
208	C₂×D₈	—	17	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	4	1	no	—
211	C₄⋊Q₈	—	34	1	no	—
212	#126	—	68	2	no	—
213	#125	—	2	2	no	—
214	#125	—	4	2	no	—
215	#124	—	34	2	no	—
216	C₂²⋊Q₈	—	68	1	no	—
217	C₄×Q₈	—	2	1	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	4	1	no	—
220	C₄×Q₈	—	34	2	no	—
221	C₄×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
222	C₄×D₄	—	34	1	no	—
223	#117	—	68	1	no	—
224	Q₃₂	—	34	4	no	—
225	SD₃₂	—	34	2	no	—
226	C₂×C₁₆	—	17	1	no	—
227	C₂×C₁₆	—	17	2	no	—
228	C₂×C₁₆	—	17	1	no	—
229	#107	—	2	1	no	—
230	#107	—	17	1	no	—
231	#107	C₂²×D₁₇	1	2	yes	—
232	#107	C₂²×D₁₇	1	1	yes	characteristic
233	#106	—	2	2	no	—
234	C₄⋊C₈	C₂²⋊D₁₇	1	1	yes	characteristic
235	C₄⋊C₈	C₂²⋊D₁₇	1	2	yes	—
236	#103	—	2	2	no	—
237	#103	—	2	1	no	—
238	#103	—	34	2	no	—
239	#102	—	34	1	no	—
240	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₁₇	1	2	yes	—
242	C₁₇	#33931	1	1	yes	characteristic, complemented, Sylow
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	34	2	no	—
246	C₂×Q₈	#63228	1	1	yes	characteristic
247	C₂×D₄	—	34	1	no	—
248	C₂²×C₄	—	68	1	no	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	34	2	no	—
251	Q₁₆	—	34	2	no	—
252	Q₁₆	—	136	1	no	—
253	Q₁₆	#63256	1	2	yes	—
254	SD₁₆	—	136	2	no	—
255	D₈	—	34	1	no	—
256	D₈	—	34	1	no	—
257	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
259	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	34	1	no	—
268	C₄⋊C₄	—	68	1	no	—
269	C₄⋊C₄	—	68	2	no	—
270	C₄⋊C₄	—	68	1	no	—
271	C₄⋊C₄	#63228	1	1	yes	characteristic
272	C₄⋊C₄	#63228	1	2	yes	—
273	C₂²⋊C₄	—	68	2	no	—
274	C₂²⋊C₄	—	68	1	no	—
275	C₂²⋊C₄	—	68	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	68	2	no	—
279	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
281	C₁₆	—	34	1	no	—
282	C₁₆	—	34	2	no	—
283	C₁₆	—	34	1	no	—
284	C₂³	—	68	1	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	8	2	no	—
289	Q₈	—	68	2	no	—
290	Q₈	—	68	2	no	—
291	D₄	—	68	1	no	—
292	D₄	—	68	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	68	1	no	—
297	C₂×C₄	—	68	2	no	—
298	C₂×C₄	—	136	2	no	—
299	C₂×C₄	C₂×#63228	1	2	yes	—
300	C₂×C₄	C₂×#63228	1	1	yes	characteristic
301	C₂×C₄	C₂×#63228	1	2	yes	—
302	C₂×C₄	C₂×#63228	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	C₂×#63256	1	1	yes	characteristic
307	C₈	#10587694	1	1	yes	characteristic
308	C₂²	—	136	1	no	—
309	C₂²	—	136	1	no	—
310	C₂²	—	136	1	no	—
311	C₂²	C₂²×#63228	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	—	136	1	no	—
319	C₄	—	136	2	no	—
320	C₄	#11763239	1	2	yes	—
321	C₄	#11763470	1	1	yes	characteristic
322	C₄	#11763617	1	1	yes	characteristic
323	C₂	—	272	1	no	—
324	C₂	#14541419	1	1	yes	characteristic
325	C₂	C₂×#11763239	1	1	yes	characteristic
326	C₂	#14541935	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#3383	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂×D₁₃₆	C₂³
hypercenter	#3383	D₁₇
hypocenter	C₁₇	#33931
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3383

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #14531013	C₂	semidirect
4. #14530945	C₂	semidirect
7. #14530834	C₂	semidirect
8. Q₈×#63214	C₂	semidirect
9. C₄×#10587500	C₂	semidirect
11. C₁₇×#3383	C₂	semidirect
146. #3383	D₁₇	semidirect
242. C₁₇	#33931	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14541773	2
#14531018	2
#14531013	2
#14530945	2
#14530918	2
#14530846	2
#14530834	2
Q₈×#63214	2
C₄×#10587500	2
C₄×#10587499	2
C₁₇×#3383	2
#33931	17